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Properties

Label 43.10.a.b
Level $43$
Weight $10$
Character orbit 43.a
Self dual yes
Analytic conductor $22.147$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,10,Mod(1,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.1465409550\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 6541 x^{15} + 10299 x^{14} + 17445509 x^{13} - 2347983 x^{12} + \cdots - 37\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3) q^{2} + ( - \beta_{3} + 10) q^{3} + (\beta_{2} - 4 \beta_1 + 267) q^{4} + ( - \beta_{9} + \beta_{3} - \beta_1 + 237) q^{5} + (\beta_{10} - \beta_{9} - 2 \beta_{3} + \cdots + 350) q^{6}+ \cdots + (\beta_{15} - \beta_{14} - 2 \beta_{10} + \cdots + 7953) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3) q^{2} + ( - \beta_{3} + 10) q^{3} + (\beta_{2} - 4 \beta_1 + 267) q^{4} + ( - \beta_{9} + \beta_{3} - \beta_1 + 237) q^{5} + (\beta_{10} - \beta_{9} - 2 \beta_{3} + \cdots + 350) q^{6}+ \cdots + ( - 30435 \beta_{16} + 2568 \beta_{15} + \cdots + 72589748) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 48 q^{2} + 169 q^{3} + 4522 q^{4} + 4033 q^{5} + 5871 q^{6} - 76 q^{7} + 41046 q^{8} + 135126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 48 q^{2} + 169 q^{3} + 4522 q^{4} + 4033 q^{5} + 5871 q^{6} - 76 q^{7} + 41046 q^{8} + 135126 q^{9} + 23763 q^{10} + 78370 q^{11} + 271339 q^{12} + 114452 q^{13} - 376208 q^{14} - 255820 q^{15} + 412586 q^{16} + 726937 q^{17} + 577055 q^{18} + 544263 q^{19} + 3642183 q^{20} + 3137394 q^{21} + 5269148 q^{22} + 5575241 q^{23} + 16215113 q^{24} + 10874708 q^{25} + 8009180 q^{26} + 8350126 q^{27} + 12534764 q^{28} + 8223345 q^{29} + 30612012 q^{30} + 13054147 q^{31} + 37111710 q^{32} + 36024808 q^{33} + 27991291 q^{34} + 17826330 q^{35} + 84105953 q^{36} + 46733879 q^{37} + 15733789 q^{38} + 8689898 q^{39} + 52241669 q^{40} + 53667013 q^{41} + 7708286 q^{42} + 58119617 q^{43} + 81727236 q^{44} + 124361968 q^{45} + 146859355 q^{46} + 122945511 q^{47} + 86356095 q^{48} + 111396073 q^{49} - 96642133 q^{50} - 187132423 q^{51} - 54447944 q^{52} - 993146 q^{53} - 219468490 q^{54} - 248155792 q^{55} - 141048116 q^{56} - 402917960 q^{57} - 466599837 q^{58} - 95519644 q^{59} - 621611940 q^{60} - 311752038 q^{61} - 212471691 q^{62} - 928966350 q^{63} - 829842590 q^{64} - 107969830 q^{65} - 978530932 q^{66} - 292438130 q^{67} - 88281129 q^{68} + 78577726 q^{69} - 1650972530 q^{70} - 13576908 q^{71} - 706943493 q^{72} - 501490738 q^{73} - 494831691 q^{74} - 641914030 q^{75} - 1248630771 q^{76} + 787365348 q^{77} - 946670550 q^{78} + 740350275 q^{79} - 27802861 q^{80} + 1582210525 q^{81} - 1600400057 q^{82} + 754109940 q^{83} - 1955423842 q^{84} + 1071609956 q^{85} + 164102448 q^{86} + 186301257 q^{87} + 1863375104 q^{88} + 1470581868 q^{89} - 698098630 q^{90} + 2895349644 q^{91} + 1041082071 q^{92} + 4540331515 q^{93} - 706582361 q^{94} + 3297255729 q^{95} + 2087289393 q^{96} + 1949310583 q^{97} + 6695989160 q^{98} + 1234191326 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 6541 x^{15} + 10299 x^{14} + 17445509 x^{13} - 2347983 x^{12} + \cdots - 37\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 770 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!87 \nu^{16} + \cdots + 62\!\cdots\!44 ) / 28\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!89 \nu^{16} + \cdots - 10\!\cdots\!60 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!69 \nu^{16} + \cdots + 11\!\cdots\!76 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 33\!\cdots\!79 \nu^{16} + \cdots - 89\!\cdots\!40 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42\!\cdots\!41 \nu^{16} + \cdots + 25\!\cdots\!00 ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!83 \nu^{16} + \cdots - 32\!\cdots\!40 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!81 \nu^{16} + \cdots - 12\!\cdots\!60 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45\!\cdots\!19 \nu^{16} + \cdots - 48\!\cdots\!60 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30\!\cdots\!53 \nu^{16} + \cdots + 99\!\cdots\!80 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!99 \nu^{16} + \cdots - 18\!\cdots\!20 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 79\!\cdots\!01 \nu^{16} + \cdots - 39\!\cdots\!80 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 31\!\cdots\!37 \nu^{16} + \cdots - 65\!\cdots\!20 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51\!\cdots\!77 \nu^{16} + \cdots - 29\!\cdots\!00 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{16} + \cdots - 49\!\cdots\!40 ) / 14\!\cdots\!20 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 770 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} + 28\beta_{3} + 2\beta_{2} + 1202\beta _1 + 1434 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{16} + 3 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 7 \beta_{12} + 5 \beta_{11} + \cdots + 934996 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{16} + 102 \beta_{15} + 50 \beta_{14} + 2 \beta_{13} - 264 \beta_{12} - 130 \beta_{11} + \cdots + 2978286 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2954 \beta_{16} + 9376 \beta_{15} - 272 \beta_{14} - 3428 \beta_{13} - 19372 \beta_{12} + \cdots + 1243629870 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 75738 \beta_{16} + 248410 \beta_{15} + 175918 \beta_{14} + 15470 \beta_{13} - 695800 \beta_{12} + \cdots + 5907635424 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4266988 \beta_{16} + 20339991 \beta_{15} - 5735802 \beta_{14} - 4914850 \beta_{13} + \cdots + 1724545573164 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 155887148 \beta_{16} + 456856200 \beta_{15} + 411347284 \beta_{14} + 54156836 \beta_{13} + \cdots + 11540559794232 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6603044108 \beta_{16} + 37986491776 \beta_{15} - 14755785840 \beta_{14} - 6722872760 \beta_{13} + \cdots + 24\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 268469732908 \beta_{16} + 765650812108 \beta_{15} + 823343670868 \beta_{14} + 126316708260 \beta_{13} + \cdots + 21\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 10857460993998 \beta_{16} + 65580253566683 \beta_{15} - 28024090051030 \beta_{14} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 408107417553334 \beta_{16} + \cdots + 40\!\cdots\!58 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 18\!\cdots\!18 \beta_{16} + \cdots + 52\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 55\!\cdots\!22 \beta_{16} + \cdots + 71\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 32\!\cdots\!24 \beta_{16} + \cdots + 77\!\cdots\!48 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
40.5953
39.1708
34.9313
33.5331
25.5994
23.8827
7.34621
1.53684
0.562738
−5.60390
−14.8584
−16.0958
−22.6959
−34.7389
−35.3457
−37.0877
−37.7321
−37.5953 −105.944 901.404 −205.302 3983.01 7435.13 −14639.7 −8458.78 7718.39
1.2 −36.1708 −228.603 796.325 2349.81 8268.76 528.884 −10284.3 32576.5 −84994.4
1.3 −31.9313 275.950 507.605 217.489 −8811.43 4698.28 140.338 56465.5 −6944.68
1.4 −30.5331 134.360 420.267 −2538.51 −4102.43 −3047.70 2800.88 −1630.30 77508.3
1.5 −22.5994 −82.1072 −1.26822 454.995 1855.57 −6480.48 11599.5 −12941.4 −10282.6
1.6 −20.8827 85.1471 −75.9135 2696.38 −1778.10 8355.76 12277.2 −12433.0 −56307.6
1.7 −4.34621 171.146 −493.110 −694.593 −743.834 −7145.50 4368.42 9607.83 3018.84
1.8 1.46316 −83.8389 −509.859 −1977.91 −122.670 −7406.23 −1495.14 −12654.0 −2894.00
1.9 2.43726 −44.8177 −506.060 1836.28 −109.232 1435.14 −2481.28 −17674.4 4475.49
1.10 8.60390 −44.7892 −437.973 −1151.27 −385.362 6348.30 −8173.47 −17676.9 −9905.41
1.11 17.8584 −263.946 −193.078 542.767 −4713.65 −12329.9 −12591.6 49984.5 9692.93
1.12 19.0958 219.847 −147.352 1171.72 4198.14 4845.77 −12590.8 28649.6 22374.8
1.13 25.6959 −189.280 148.278 −1738.18 −4863.71 2771.25 −9346.16 16143.9 −44664.0
1.14 37.7389 83.4314 912.221 1398.64 3148.61 8642.65 15103.9 −12722.2 52783.1
1.15 38.3457 −137.794 958.392 2131.30 −5283.82 −2993.43 17117.2 −695.689 81726.4
1.16 40.0877 243.316 1095.02 1211.89 9753.98 −12008.0 23372.0 39519.8 48581.8
1.17 40.7321 136.923 1147.10 −1672.50 5577.18 6274.02 25869.0 −934.956 −68124.5
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.10.a.b ✓ 17
3.b odd 2 1 387.10.a.e 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.a.b ✓ 17 1.a even 1 1 trivial
387.10.a.e 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 48 T_{2}^{16} - 5461 T_{2}^{15} + 268926 T_{2}^{14} + 11844242 T_{2}^{13} + \cdots + 17\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(43))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{17} + \cdots + 76\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots + 76\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 52\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 55\!\cdots\!66 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 91\!\cdots\!30 \) Copy content Toggle raw display
$43$ \( (T - 3418801)^{17} \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 74\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 66\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 32\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 13\!\cdots\!78 \) Copy content Toggle raw display
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